Minimized Thompson NFA

The problem of converting a regular expression to NFA is a fundamental problem that has been well studied. However, the two basic construction algorithms: (1) Thompson, (2) McNaughton-Yamada and Glushkov, both have disadvantages. In this paper: First, a “smart” parsing algorithm is developed which constructs a parse tree with at most (3l − 1) nodes form a regular expression with l literals; Second, we propose an algorithm that works on the resulting NFA from Thompson’s construction, eliminating as many auxiliary states as possible while maintaining Thompson’s properties. It is shown that the resulting NFA is minimized. This means that no auxiliary states can be eliminated without violating the defining properties of Thompson NFA. The time and space requirement for the above algorithms are linear with respect to the length of the regular expression. To the author’s knowledge, this is the first linear time algorithm minimizing an NFA in a precise technical sense.